Existence and uniqueness of well-posed fractional boundary value problem

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer’s fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.


Introduction
The specific problem and the constraints placed on it determine if there are solutions to fractional boundary value problems and whether they are unique.I can, however, provide you some general information regarding the well-posedness of problems involving fractional boundary values.
Differential equations with fractional derivatives are used in fractional boundary value problems.The fractional derivative of a function is a non-integer order extension of the classical derivative.Modelling phenomena exhibiting non-local behaviour and long-range interactions frequently makes use of these kinds of issues [1,2].
An explanation of the issue A fractional differential equation and boundary conditions that specify how the solution should behave at the domain's edges are often used to solve a fractional boundary value problem.The issue must be clearly stated and be sound mathematically.
One must take into account various factors in order to judge whether a fractional boundary value problem is well-posed: Existence of solutions: Several techniques, including fixed-point theorems, variational methods, and semi group theory, can be used to prove the existence of solutions.The particular situation at hand and the relevant fractional operator's characteristics determine the method to be used.
Uniqueness of solutions: Compared to classical differential equations, fractional boundary value issues frequently provide more difficulties in proving the uniqueness of solutions.It can call for further presumptions or certain characteristics of the situation.Under the right circumstances, a variety of methods, such as integral equations, fractional calculus, or comparison principles, can be employed to demonstrate uniqueness.
Stability and regularity: Another aspect of well-posedness is the solution's consistency and stability.Regularity is concerned with the solution's smoothness or differentiability, whereas stability is concerned with the solution's sensitivity to changes in the initial or boundary data.
It is significant to highlight that research on the well-posedness of fractional boundary value problems is ongoing, and the outcomes depend on the particular problem and the fractional order at play.In comparison to classical calculus, the area of fractional calculus is still developing and many elements are still being researched.
In a wide range of mathematical problems, the existence of a solution is equivalent to the existence of a fixed point for a suitable map.The existence of a fixed point is therefore of paramount importance in several areas of mathematics and other sciences.Existence and uniqueness theorems are useful to find one or more conditions which show that there is exactly one solution to given problem.The existence and uniqueness is proved using fixed point theorems which means that the given fractional differential equations has one fixed point under some suitable conditions on fractional differential equations.Fixed points are useful because many problems in mathematics can be formulated in terms of the existence of a fixed point, and it's often much easier to show that such points exist and then to approximate them (e.g.numerically) than it is to actually find them explicitly.
Fractional differential equations have more importance in practical field due to its applications in science and engineering such as electrochemistry, fluid flow, rheology, diffusive transport, electrical networks, probability, viscoelasticity, control image, signal processing, biophysics, and electromagnetic theory etc [3][4][5][6][7][8][9].Many researchers worked on the applications of fractional differential equations such as Rousan et al. [10] suggested a fractional differential equation that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit.The concept of fractional derivatives is employed in the formulation of a stress-stain relationship for elastomers by Koh and Kelly [11].They also developed numerical schemes for the dynamic analysis of a single degree of freedom fractional oscillator in the time domain.From last few years existence and uniqueness of boundary value problem (BVP) of corresponding FDEs have been attracted the interest of many researchers.Many mathematicians such as Ahmad et al. worked on existence of Caputo type fractional differential equations with four-point non-local fractional integral boundary conditions [12].Benchohra and Ouaar [13] investigated the existence of solutions for FDEs with integral conditions.Ntouyas worked on existence of first order boundary value problems for fractional differential equations and inclusions with fractional integral boundary condition (BC) [14].Chai established the existence results of positive solutions for boundary value problems of fractional differential equations [15].
Ahmad and Nieto studied model containing R-L(Riemann-Liouville) fractional integro-differential equations with fractional nonlocal integral BC [16].Zhong and Lin [17] established the solvability results for a FDM (fractional differential model) having nonlocal and multipoint BVP for FDEs of order 1 < γ � 2. Lakoud and Khaldi [18] established sufficient conditions for the existence and uniqueness of solutions of fractional differential equations with fractional integral condition, involving the Caputo fractional derivative.Choudhary and Daftar [19] established existence and uniqueness results on fractional differential model which contains a non-linear multi-order FDE and periodic and anti-periodic boundary conditions.They used green function for finding their results.[20] presented the existence and uniqueness of three point BVP of noninteger order β 2 [2,3] which includes M-L (Mittag-Lefer) function.They found the solution of considered BVP in term of M-L function using Caputo fraction derivative then check the existence and uniqueness of solution using fixed point theorems.
This paper contains a fractional differential model which involves an implicit FDE and well-posed boundary conditions such as: where, γ, z are the order of Caputo fractional derivatives, t 2 [0, 1], 0 < x < 1 m 2 and 0 < l < 1 n 2 .λ, ξ, ν, μ these are parameters on which boundary conditions depends.J z is an Riemann-Liouville fractional integral operator of order z.And C γ and C z are Caputo derivatives.We will use Caputo derivative due to its main advantage which is that the initial and boundary conditions for differential equations with the Caputo fractional derivative are analogous to the case of integer order differential equations, so they can be interpreted in the same way.Therefore, it is often used in practical applications.
Houas and benbachir worked for the fractional differential model involving fractional order α 2 (2, 3] but they did not discuss about the the fractional differential model of order (3,4], this paper will fill this gab. The solution of a fractional differential equation depends upon different types boundary conditions or initial conditions.Our boundary conditions are well-posed which mean the solution behaviour changes continuously with boundary condition and solution is not highly sensitive to changes in final data.Solution of well-posed boundary condition problem can be deducted on a computer using stable algorithm.On the other hand for ill-possed (not wellposed) problem one have to reformulate the algorithm for numerical analysis.There are lot of applications of the fourth order ordinary differential problem in literature such as nonlinear models of suspension bridge [40].Our model help to convert such type of forth order differential problems in fractional differential model so that the problem will became more accurate as compared to the ordinary differential problem.
This paper is organized as follows: In Section 3, some useful definitions and Lemmas of fractional calculus are presented.The general solution of four-point BVP is described in Section 4. In Section 5, the existence of BVP (1) is proved by Scheafer fixed point theorem.In the same Section uniqueness of BVP (1) is proved by Banach contraction principle.

General solution of BVP (1)
Observed the following BVP as: where, L on interval [0, 1] is a absolutely continuous function.

Existence and uniqueness
Some important notations are as follows: List of hypothesis for this paper are as follows: (H1): Suppose g : IR 2 !IR is a continuous function.
(H2): Suppose that there exists non negative real numbers θ, ϑ > 0 such that for all ðu; vÞ; (H3): Suppose that a positive real number M exists such that |g(t, u, v)| � M for all u, v 2 X and for each t 2 [0, 1].
Proof: The proof of this theorem will be given in four steps.
Step 1: In X the operator τ is continuous.
It is obvious that the operator τ is continuous because the function g is continuous function.
For u 2 B ω and t 2 [0, 1], implies By (H3), the above relation can be express as Consequently, The above expression can be written as: Using Eqs 27 and 30 gives, Step 3: In this step we will prove that the operator τ is equi-continuous on J: Assume that if t 1 , t 2 where t 2 > t 1 and u 2 B ω ., then, Using (H3), and, ð36Þ which shows that kτu(t 2 ) − τu(t 1 )k X !0 if t 2 !t 1 , then by Arzela-Ascoli theorem it is concluded that τ is completely continuous operator.
This implies to Hence, by Theorem (4.1), the BVP (1) has a unique solution.
Example 2: Consider the following boundary value problem: Step 1: It is obvious that the operator τ is continuous because the function |g(u, v)| = | 2t 2 cosu + tsinv| is continuous function.
It is clear from the above equation that every term has t 2 − t 1 so if t 2 !t 1 then kτu(t 2 ) − τu (t 1 )k X !0. By Arzela-Ascoli theorem it is concluded that τ equi-continuous operator.

Conclusion
We managed to employ Banach contraction principle and Schaefer fixed point theorem to study the existence and uniqueness of solution for a well-possed implicit fractional boundary value problem involving Caputo derivatives of order γ 2 [3,4] and z 2 [1,2].Results of this paper give a different opinion about the differentiation of derivative in the discussed boundary value problem.Considered function is necessarily third-order differentiable to guarantee the existence of solution could be forth order differentiable.If the function is not third order differentiable then the solution u(t) is not exist for q = 4 because d 4 /dx 4 does not exist.
The model under study is generalized version of many recent studies.We use some examples to demonstrate the results.In future this model will help to convert forth order differential problems in fractional differential models.The main advantage of this conversion is fractional differential models are more accurate as compare to ordinary differential model because it preserves the history of procedure and predict the future.